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Progress in Interleaved Manipulation

Following on our work in the 2013 International Conference on Robotics and Automation, we continued exploring interleaved continuum-rigid manipulation in the journal of Robotics and Autonomous Systems. We explored a number of new facets of this concept including evaluation of a new controller, the effects of rigid actuator saturation, the manipulation range overlap of the rigid and flexible joints, and increase in dexterity from the addition of the rigid joint.

Controller Improvement

The controller used in this paper is similar to that presented at ICRA 2013, but different from that described in the ICRA proceedings paper. This controller was inspired by a controller for hard drives where the position error is partitioned between fast and slow actuators. Applied to our interleaved continuum-rigid catheter experiment, this frequency partitioning takes the form:

Overview of a candidate interleaved continuum-rigid manipulator control scheme

The particular Df and Dr compensator design and tuning determines what error is directed to the flexible and rigid actuators, leaving a system that can be treated as if it has a single actuator. The Dl loop compensator then tunes this 'single system' to the desired performance, as follows:

Closed-loop frequency response of a one degree-of-freedom simplified interleaved system. Left - magnitude and phase as a function of frequency, where the contribution of the flexible segment articulation, rigid-link actuation, and the combination of the two are shown. Right - Frequency response of the system.

Effects of Rigid Actuator Saturation

While the actuation ranges of the rigid and flexible actuators must overlap for the rigid actuator to be able to compensate for errors in the flexible actuator, the rigid and flexible actuators will have different actuation ranges. For example, in the experimental testbed the rigid actuator has ~10° of actuation range, while the flexible segment can articulate nearly 360°. If the rigid link reaches the end of its actuation range (that is it saturates), it will not be able to achieve the positions commanded it, leading to the following aberrant behavior:

We show that as the saturation increases the phase margin decreases quickly, leading to the oscillatory behavior seen above.

Error Bounds

Since the rigid and flexible joints are different, we cannot always expect that the joint-space errors for each actuator will ideally overlap in task-space. Thus depending on the particular characteristics of the flexible and rigid joints and the manipulator configuration, there may be some flexible segment errors which the rigid link actuator cannot compensate:

Flexible segment and rigid-link task-space error and motion bounds for a two degree of freedom manipulator.

Dexterity Increase

The addition of the rigid link joint has the potential to beneficially increase the overall workspace of the manipulator:

Comparison of the dexterous workspaces for a simple three degree of freedom manipulator with a horizonal tip task constraint. Left - Workspace of a three segment manipulator. Right - Workspace of a two segement, one rigid link manipulator.

In this notional example, the tip is required to always point horizontally. Despite both manipulators having three degrees of freedom, the different character of the rigid link joint substantially increases the workspace. So in addition to correcting for flexible segment errors, the rigid link joint can also greatly increase workspace, limited only by the particular rigid link joint design.